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We establish a new $W^{1,2\frac{n-1}{n-2}}$ estimate for the extremalsolution of $-\Delta u=\lambda f(u)$ in a smooth bounded domain $\Omega$ of$\mathbb{R}^n$, which is convex, for arbitrary positive and increasingnonlinearities $f\in C^1(\mathbb{R})$ satisfying $\lim_{t\rightarrow+\infty}f(t)/t=+\infty$.

$W^{1,q}$ estimates for the extremal solution of reaction-diffusion problems